Determinants and Invertibility

The determinant of a $2\times 2$ matrix $\begin{pmatrix}a&b\\c&d\end{pmatrix}$ is the single number

It reveals whether the matrix can be inverted (undone). If $\det \neq 0$ the matrix is invertible; if $\det = 0$ it is singular and has no inverse.

When it exists, the inverse of $\begin{pmatrix}a&b\\c&d\end{pmatrix}$ is

The determinant sits in the denominator — which is exactly why a zero determinant blocks inversion (you cannot divide by zero).

Worked example. $\det\begin{pmatrix}3&1\\2&4\end{pmatrix} = 3\cdot 4 - 1\cdot 2 = 12 - 2 = 10 \neq 0$, so this matrix is invertible.